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import FormalConjecturesUtilExpansion of g.f. $-(1 - 48x^2 - 256x^3) / ((1 - 4x)(1 + 4x)(1 + 4x + 16x^2))$
Corresponds to m = 4 in a family of 4th-order linear recurrence sequences
This sequence is defined by the linear recurrence relation with signature $(-4, 0, 64, 256)$ and initial values $a(0) = -1, a(1) = 4, a(2) = 32, a(3) = 64$. The recurrence is $a(n) = -4 a(n-1) + 64 a(n-3) + 256 a(n-4)$.
References:
namespace OeisA113250a n is the sequence defined by the linear recurrence $a(n) = -4 a(n-1) + 64 a(n-3) + 256 a(n-4)$ with initial values $-1, 4, 32, 64$.
def a : ℕ → ℤ
| 0 => -1
| 1 => 4
| 2 => 32
| 3 => 64
| n + 4 => -4 * a (n + 3) + 64 * a (n + 1) + 256 * a n@[category test, AMS 11]
theorem a_0 : a 0 = -1 := ⊢ a 0 = -1 All goals completed! 🐙@[category test, AMS 11]
theorem a_1 : a 1 = 4 := ⊢ a 1 = 4 All goals completed! 🐙@[category test, AMS 11]
theorem a_2 : a 2 = 32 := ⊢ a 2 = 32 All goals completed! 🐙@[category test, AMS 11]
theorem a_3 : a 3 = 64 := ⊢ a 3 = 64 All goals completed! 🐙@[category test, AMS 11]
theorem a_4 : a 4 = -256 := ⊢ a 4 = -256 All goals completed! 🐙Conjecture: $a(m, 2n+1)$ is a perfect square for all $m$ (see A113249). Specialized to $m = 4$, which is A113250.
@[category research solved, AMS 11,
formal_proof using lean4 at
"https://github.com/KitaKen1/oeis-a113249-family-square-terms-lean/blob/9b999db08344184285e8050c2722c845fd5f5309/lean/OeisA113249FamilyFC.lean#L106-L114"]
theorem conjecture : ∀ n : ℕ, IsSquare (a (2 * n + 1)) := ⊢ ∀ (n : ℕ), IsSquare (a (2 * n + 1))
All goals completed! 🐙end OeisA113250